Axiomatic method of partitions in the theory of Nöbeling spaces. I. Improvement of partition connectivity

The Nöbeling space $N_k^{2k+1}$, a $k$-dimensional analogue of the Hilbert space, is considered; this is a topologically complete separable (that is, Polish) $k$-dimensional absolute extensor in dimension $k$ (that is, $\mathrm{AE}(k)$) and a strongly $k$-universal space. The conjecture that the above-listed properties characterize the Nöbeling space $N_k^{2k+1}$ in an arbitrary finite dimension $k$ is proved. In the first part of the paper a full axiom system of the Nöbeling spaces is presented and the problem of the improvement of a partition connectivity is solved on its basis.
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